Blue's Berry Farm charges Percy a total of $\$24.75$ for entrance and $2.5$ kilograms of strawberries. The entrance fee is $\$6$, and the price for each kilogram of strawberries is constant. The total cost $C$ for picking strawberries is a function of $x$, the kilograms of strawberries picked. Write the function's formula. $C=$
Solution: The farm charges a constant rate for each kilogram of strawberries, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $C= mx+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that the entrance fee is $\$6$, so the $y$ -intercept ${b}$ is ${6}$, and our function looks like $C={m}x+{6}$. We also know that the farm charges Percy $\$24.75$ for $2.5$ kilograms of strawberries, which means when $x=2.5$, $C=24.75$. We can use this and the $y$ -intercept to find ${m}$ : $\begin{aligned} {m}&=\dfrac{C_2-C_1}{x_2-x_1} \\\\ &=\dfrac{24.75-6}{2.5-0} \\\\ &=\dfrac{18.75}{2.5} \\\\ &={7.5} \end{aligned}$ This means the farm charges $\$7.50$ for each kilogram of strawberries. Since ${m}={7.5}$ and ${b}={6}$, the desired formula is: $C={7.5} x + {6}$